module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.FieldTheory.Differential.Basic | {
"line": 179,
"column": 6
} | {
"line": 179,
"column": 63
} | [
{
"pp": "case a.deriv.H.inj.ha.h0\nR : Type u_1\ninst✝⁷ : Field R\ninst✝⁶ : Differential R\na✝ b : R\nF : Type u_2\ninst✝⁵ : Field F\ninst✝⁴ : Differential F\ninst✝³ : CharZero F\nK : Type u_3\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : FiniteDimensional F K\ndefault : { _a // DifferentialAlgebra F K } := ... | simp [natDegree_eq_zero_of_derivative_eq_zero nh] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Discriminant | {
"line": 283,
"column": 28
} | {
"line": 283,
"column": 41
} | [
{
"pp": "K : Type u\nL : Type v\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : Module.Finite K L\nR : Type z\ninst✝⁶ : CommRing R\ninst✝⁵ : Algebra R K\ninst✝⁴ : Algebra R L\ninst✝³ : IsScalarTower R K L\ninst✝² : Algebra.IsSeparable K L\ninst✝¹ : IsIntegrallyClosed R\ninst✝ : IsFractionRi... | cramer_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Invariant.Basic | {
"line": 143,
"column": 81
} | {
"line": 144,
"column": 51
} | [
{
"pp": "B : Type u_2\nG : Type u_3\ninst✝³ : CommRing B\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G B\ninst✝ : Fintype G\nb : B\n⊢ charpoly G b = ∏ g, g • (X - C b)",
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"Finset.univ",
... | by
simp only [smul_sub, smul_C, smul_X, charpoly_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Galois.NormalBasis | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 76
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : FiniteDimensional K L\ninst✝ : Infinite K\ne : Module.Basis (Module.Free.ChooseBasisIndex K L) K L\nM : Matrix Gal(L/K) Gal(L/K) (MvPolynomial (Module.Free.ChooseBasisIndex K L) L) :=\n Matrix.of fun i j ↦ ∑... | refine Matrix.eq_zero_of_mulVec_eq_zero hb (funext fun i ↦ i.injective ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Invariant.Basic | {
"line": 324,
"column": 45
} | {
"line": 324,
"column": 64
} | [
{
"pp": "case intro\nA : Type u_1\nB : Type u_2\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : CommRing B\ninst✝¹⁶ : Algebra A B\nG : Type u_3\ninst✝¹⁵ : Group G\ninst✝¹⁴ : Finite G\ninst✝¹³ : MulSemiringAction G B\ninst✝¹² : SMulCommClass G A B\nP : Ideal A\nQ : Ideal B\ninst✝¹¹ : Q.IsPrime\ninst✝¹⁰ : Q.LiesOver P\nK : Type... | Polynomial.map_prod | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic | {
"line": 318,
"column": 26
} | {
"line": 318,
"column": 74
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ ProfiniteGrp.{max v u}\na✝ b✝ : (j : J) → ↑(F.obj j).toProfinite.toTop\nhx : a✝ ∈ {x | ∀ ⦃i j : J⦄ (π : i ⟶ j), (Hom.hom (F.map π)) (x i) = x j}\nhy : b✝ ∈ {x | ∀ ⦃i j : J⦄ (π : i ⟶ j), (Hom.hom (F.map π)) (x i) = x j}\nx✝¹ x✝ : J\nπ : x✝¹ ⟶ x✝\n⊢ (Hom.hom (... | by simp only [Pi.mul_apply, map_mul, hx π, hy π] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Invariant.Basic | {
"line": 517,
"column": 63
} | {
"line": 517,
"column": 82
} | [
{
"pp": "case inr.splits'.intro.refine_1\nA : Type u_1\nB : Type u_2\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\nG : Type u_4\ninst✝⁶ : Finite G\ninst✝⁵ : Group G\ninst✝⁴ : MulSemiringAction G B\ninst✝³ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝² : Q.LiesOver P\ninst✝¹ : P... | Polynomial.map_prod | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IsPerfectClosure | {
"line": 158,
"column": 51
} | {
"line": 165,
"column": 64
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : CommSemiring K\ninst✝¹ : CommSemiring L\ni : K →+* L\np : ℕ\ninst✝ : IsPRadical i p\n⊢ Ideal.comap i (pNilradical L p) = pNilradical K p",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submodule",
"RingHom.instRingHomClass",
"_private.Ma... | by
refine le_antisymm (fun x h ↦ mem_pNilradical.2 ?_) (fun x h ↦ ?_)
· obtain ⟨n, h⟩ := mem_pNilradical.1 <| Ideal.mem_comap.1 h
obtain ⟨m, h⟩ := mem_pNilradical.1 <| ker_le i p ((map_pow i x _).symm ▸ h)
exact ⟨n + m, by rwa [pow_add, pow_mul]⟩
simp only [Ideal.mem_comap, mem_pNilradical] at h ⊢
obtai... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.IsPerfectClosure | {
"line": 260,
"column": 67
} | {
"line": 264,
"column": 52
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁴ : CommRing K\ninst✝³ : CommRing L\ninst✝² : CommRing M\ni : K →+* L\np : ℕ\ninst✝¹ : ExpChar M p\ninst✝ : IsPRadical i p\nh : pNilradical M p = ⊥\nf g : L →+* M\nheq : (fun f ↦ f.comp i) f = (fun f ↦ f.comp i) g\n⊢ f = g",
"usedConstants": [
"p... | by
ext x
obtain ⟨n, y, hx⟩ := IsPRadical.pow_mem i p x
apply_fun _ using pow_expChar_pow_inj_of_pNilradical_eq_bot M p h n
simpa only [← map_pow, ← hx] using congr($(heq) y) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.IsPerfectClosure | {
"line": 524,
"column": 4
} | {
"line": 524,
"column": 34
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : K\nn : ℕ\nh : (iterateFrobenius K p n) x = 0\n⊢ x ∈ pNilradical K p",
"usedConstants": [
"Iff.mpr",
"Semiring.toModule",
"CommSemiring.toSemiring... | exact mem_pNilradical.2 ⟨n, h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Isaacs | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 63
} | [
{
"pp": "case inl\nF : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nalg : Algebra.IsAlgebraic F E\nh : ∀ (x : E), ∃ y, (aeval y) (minpoly F x) = 0\nS : Finset E\np : K[X] := ∏ x ∈ S, Polynomial.map (algebraMap F K) (minpol... | have ⟨ω, hω⟩ := Set.mem_iUnion.mp (this ▸ Set.mem_univ α) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.JacobsonNoether | {
"line": 128,
"column": 37
} | {
"line": 128,
"column": 76
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Algebra.IsAlgebraic (↥k) D\nH : k ≠ ⊤\np : ℕ\nhp : ExpChar D p\ninsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k\na : D\nha : a ∉ k\nha₀ : a ≠ 0\nb : D\nhb1 : ((ad (↥k) D) a) b ≠ 0\nm : ℕ\nhm2 : ∀ (n : ℕ), p ^ m ≤ n → (⇑((ad (↥k) D) a))^[n] = 0\n⊢ (⇑((ad (↥k)... | hm2 (p ^ m + 1) (Nat.le_add_right _ _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality | {
"line": 59,
"column": 74
} | {
"line": 59,
"column": 85
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nι : Type w\nx : ι → E\ninst✝ : Nonempty ι\nhx : IsTranscendenceBasis F x\nK : IntermediateField F E := IntermediateField.adjoin F (range x)\nthis : Algebra.IsAlgebraic (↥K) E\n⊢ max (max (lift.{max w v, u} #F) (lift.{max ... | lift_aleph0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 325,
"column": 77
} | {
"line": 325,
"column": 86
} | [
{
"pp": "K : Type u\ninst✝³ : Field K\nn : ℕ\nhζ✝ : (primitiveRoots n K).Nonempty\na : K\nH✝ : Irreducible (X ^ n - C a)\nL : Type u_1\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsSplittingField K L (X ^ n - C a)\nα : L\nhα✝ : α ^ n = (algebraMap K L) a\nhζ : (primitiveRoots n K).Nonempty\nH : Irreducible... | simp [hα] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.KummerExtension | {
"line": 325,
"column": 77
} | {
"line": 325,
"column": 86
} | [
{
"pp": "K : Type u\ninst✝³ : Field K\nn : ℕ\nhζ✝ : (primitiveRoots n K).Nonempty\na : K\nH✝ : Irreducible (X ^ n - C a)\nL : Type u_1\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsSplittingField K L (X ^ n - C a)\nα : L\nhα✝ : α ^ n = (algebraMap K L) a\nhζ : (primitiveRoots n K).Nonempty\nH : Irreducible... | simp [hα] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.KummerExtension | {
"line": 325,
"column": 77
} | {
"line": 325,
"column": 86
} | [
{
"pp": "K : Type u\ninst✝³ : Field K\nn : ℕ\nhζ✝ : (primitiveRoots n K).Nonempty\na : K\nH✝ : Irreducible (X ^ n - C a)\nL : Type u_1\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsSplittingField K L (X ^ n - C a)\nα : L\nhα✝ : α ^ n = (algebraMap K L) a\nhζ : (primitiveRoots n K).Nonempty\nH : Irreducible... | simp [hα] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.KummerExtension | {
"line": 388,
"column": 2
} | {
"line": 388,
"column": 54
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\nn : ℕ\nhζ : (primitiveRoots n K).Nonempty\na : K\nH : Irreducible (X ^ n - C a)\nL : Type u_1\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : IsSplittingField K L (X ^ n - C a)\ninst✝ : NeZero n\nη : ↥(rootsOfUnity n K)\n⊢ ((autEquivRootsOfUnity hζ H L).symm η) (rootOfSp... | rw [MulEquiv.apply_symm_apply, autEquivRootsOfUnity] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.KummerExtension | {
"line": 397,
"column": 47
} | {
"line": 407,
"column": 23
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\nn : ℕ\nhζ : (primitiveRoots n K).Nonempty\na : K\nH : Irreducible (X ^ n - C a)\nL : Type u_1\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : IsSplittingField K L (X ^ n - C a)\nα : L\nhα : α ^ n = (algebraMap K L) a\ninst✝ : NeZero n\nσ : Gal(L/K)\n⊢ (autEquivRootsOfUni... | by
have ⟨ζ, hζ'⟩ := hζ
have hn := NeZero.pos n
rw [mem_primitiveRoots hn] at hζ'
rw [← mem_nthRoots hn, (hζ'.map_of_injective (algebraMap K L).injective).nthRoots_eq
(rootOfSplitsXPowSubC_pow a L)] at hα
simp only [Multiset.mem_map, Multiset.mem_range] at hα
obtain ⟨i, _, rfl⟩ := hα
simp only [← map_p... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Minpoly.ConjRootClass | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 61
} | [
{
"pp": "case h\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsAlgebraic K L\nx✝ : L\n⊢ Irreducible (minpoly K x✝)",
"usedConstants": [
"Algebra.IsIntegral.isIntegral",
"Field.toDivisionRing",
"DivisionRing.toRing",
"Field.toS... | exact minpoly.irreducible (Algebra.IsIntegral.isIntegral _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.RatFunc.Degree | {
"line": 47,
"column": 53
} | {
"line": 48,
"column": 79
} | [
{
"pp": "K : Type u\ninst✝ : Field K\n⊢ intDegree 0 = 0",
"usedConstants": [
"Eq.mpr",
"Polynomial.instOne",
"RatFunc.denom",
"sub_self",
"congrArg",
"AddMonoid.toAddZeroClass",
"HSub.hSub",
"AddZeroClass.toAddZero",
"id",
"RatFunc.num_zero",
... | by
rw [intDegree, num_zero, natDegree_zero, denom_zero, natDegree_one, sub_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PurelyInseparable.Tower | {
"line": 116,
"column": 2
} | {
"line": 117,
"column": 59
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁸ : Field F\ninst✝⁷ : Field E\ninst✝⁶ : Algebra F E\nK : Type w\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\ninst✝³ : Algebra E K\ninst✝² : IsScalarTower F E K\ninst✝¹ : IsPurelyInseparable F E\ninst✝ : Algebra.IsSeparable E K\n⊢ sepDegree F K = Module.rank E K",
"usedConst... | have h := (separableClosure F K).linearDisjoint_of_isPurelyInseparable_of_isSeparable E
|>.adjoin_rank_eq_rank_left_of_isAlgebraic_left |>.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.PurelyInseparable.Tower | {
"line": 196,
"column": 7
} | {
"line": 196,
"column": 65
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁷ : Field F\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nK : Type w\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower F E K\ninst✝ : Algebra.IsAlgebraic F E\nh :\n Cardinal.lift.{w, v} (Module.rank (↥(separableClosure F E)) E) * Cardinal.lif... | ← insepDegree_eq_of_isSeparable F (separableClosure F E) K | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Relrank | {
"line": 245,
"column": 2
} | {
"line": 246,
"column": 61
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B C : Subfield E\n⊢ A.relrank (B ⊓ C) * B.relrank C = (A ⊓ B).relrank C",
"usedConstants": [
"Eq.mpr",
"Subfield.relrank",
"HMul.hMul",
"Cardinal",
"congrArg",
"Field.toDivisionRing",
"inf_le_right",
"Subfield.instMin",
... | rw [← inf_relrank_right A (B ⊓ C), ← inf_relrank_right B C, ← inf_relrank_right (A ⊓ B) C,
inf_assoc, relrank_mul_relrank inf_le_right inf_le_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Relrank | {
"line": 245,
"column": 2
} | {
"line": 246,
"column": 61
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B C : Subfield E\n⊢ A.relrank (B ⊓ C) * B.relrank C = (A ⊓ B).relrank C",
"usedConstants": [
"Eq.mpr",
"Subfield.relrank",
"HMul.hMul",
"Cardinal",
"congrArg",
"Field.toDivisionRing",
"inf_le_right",
"Subfield.instMin",
... | rw [← inf_relrank_right A (B ⊓ C), ← inf_relrank_right B C, ← inf_relrank_right (A ⊓ B) C,
inf_assoc, relrank_mul_relrank inf_le_right inf_le_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Relrank | {
"line": 245,
"column": 2
} | {
"line": 246,
"column": 61
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B C : Subfield E\n⊢ A.relrank (B ⊓ C) * B.relrank C = (A ⊓ B).relrank C",
"usedConstants": [
"Eq.mpr",
"Subfield.relrank",
"HMul.hMul",
"Cardinal",
"congrArg",
"Field.toDivisionRing",
"inf_le_right",
"Subfield.instMin",
... | rw [← inf_relrank_right A (B ⊓ C), ← inf_relrank_right B C, ← inf_relrank_right (A ⊓ B) C,
inf_assoc, relrank_mul_relrank inf_le_right inf_le_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.Luroth | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 65
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\nthis :\n (algebraMap K[X] K⟮X⟯) ((Φ E).coeff (generatorIndex h) * g E) = (algebraMap K[X] K⟮X⟯) ((c E).num * (generator E).num)\n⊢ (f E).natDegree ≤ ((Φ E).coeff (generatorIndex h)).natDegree",
"usedConstants": [
"HMul.h... | replace this := congr($(algebraMap_injective K this).natDegree) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 158,
"column": 6
} | {
"line": 159,
"column": 98
} | [
{
"pp": "case mp.right\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\np : P\nh : vectorSpan ℝ {p, s.points i} = (s.altitude i).direction\n⊢ p -ᵥ s.point... | exact
vsub_mem_vectorSpan ℝ (Set.mem_insert _ _) (Set.mem_insert_of_mem _ (Set.mem_singleton _)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Projection | {
"line": 121,
"column": 5
} | {
"line": 123,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np : P\n⊢ IsCompl s.direction (... | by
rw [direction_mk' p s.directionᗮ]
exact Submodule.isCompl_orthogonal_of_hasOrthogonalProjection | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 24
} | [
{
"pp": "case neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh₃₂ : p₃ ≠ p₂\n⊢ ∠ p₁ p₂ ((AffineEquiv.pointReflection ℝ p₂) p₃) = π - ∠ p₁ p₂ p₃",
"usedConstants": [
"Eq.mpr",
"Inner... | rw [eq_sub_iff_add_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Projection | {
"line": 284,
"column": 6
} | {
"line": 284,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace 𝕜 V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace 𝕜 P\ninst✝³ : Nonempty ↥s₁\ninst✝² : Nonempty ↥s₂\ninst✝¹ : s₁.direction.HasOrthogonalProjection\n... | orthogonalProjection_eq_orthogonalProjection_iff_vsub_mem | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 112,
"column": 46
} | {
"line": 112,
"column": 71
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫ = 0\n⊢ Real.arccos (‖x‖ / ‖x + y‖) ≤ π / 2",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"Real.pi",
"Real.instZero",
"cong... | Real.arccos_le_pi_div_two | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 370,
"column": 2
} | {
"line": 378,
"column": 91
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\n⊢ o.oangle x y = ↑π ↔ x ≠ 0 ∧ y ≠ 0 ∧ SameRay ℝ x (-y)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"InnerProductSpa... | rw [← o.oangle_eq_zero_iff_sameRay]
constructor
· intro h
by_cases hx : x = 0; · simp [hx, Real.Angle.pi_ne_zero.symm] at h
by_cases hy : y = 0; · simp [hy, Real.Angle.pi_ne_zero.symm] at h
refine ⟨hx, hy, ?_⟩
rw [o.oangle_neg_right hx hy, h, Real.Angle.coe_pi_add_coe_pi]
· rintro ⟨hx, hy, h⟩
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 370,
"column": 2
} | {
"line": 378,
"column": 91
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\n⊢ o.oangle x y = ↑π ↔ x ≠ 0 ∧ y ≠ 0 ∧ SameRay ℝ x (-y)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"InnerProductSpa... | rw [← o.oangle_eq_zero_iff_sameRay]
constructor
· intro h
by_cases hx : x = 0; · simp [hx, Real.Angle.pi_ne_zero.symm] at h
by_cases hy : y = 0; · simp [hy, Real.Angle.pi_ne_zero.symm] at h
refine ⟨hx, hy, ?_⟩
rw [o.oangle_neg_right hx hy, h, Real.Angle.coe_pi_add_coe_pi]
· rintro ⟨hx, hy, h⟩
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 365,
"column": 2
} | {
"line": 367,
"column": 54
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nh : o.oangle x y = ↑(π / 2)\nhs : (o.oangle y (y - x)).sign = 1\n⊢ (o.oangle y (y - x)).sin * ‖y - x‖ = ‖x‖",
"usedConstants": [
"Norm.norm",
"... | rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
InnerProductGeometry.sin_angle_sub_mul_norm_of_inner_eq_zero
(o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 560,
"column": 35
} | {
"line": 560,
"column": 88
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₃' : P\nh : Wbtw ℝ p₂ p₃ p₃'\nhp₃p₂ : p₃ ≠ p₂\n⊢ ∡ p₁ p₂ p₃ = ∡ p₁ p₂ p₃'",
"usedCon... | rw [oangle_rev, h.oangle_eq_left hp₃p₂, ← oangle_rev] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 560,
"column": 35
} | {
"line": 560,
"column": 88
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₃' : P\nh : Wbtw ℝ p₂ p₃ p₃'\nhp₃p₂ : p₃ ≠ p₂\n⊢ ∡ p₁ p₂ p₃ = ∡ p₁ p₂ p₃'",
"usedCon... | rw [oangle_rev, h.oangle_eq_left hp₃p₂, ← oangle_rev] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 560,
"column": 35
} | {
"line": 560,
"column": 88
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₃' : P\nh : Wbtw ℝ p₂ p₃ p₃'\nhp₃p₂ : p₃ ≠ p₂\n⊢ ∡ p₁ p₂ p₃ = ∡ p₁ p₂ p₃'",
"usedCon... | rw [oangle_rev, h.oangle_eq_left hp₃p₂, ← oangle_rev] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 636,
"column": 4
} | {
"line": 636,
"column": 89
} | [
{
"pp": "case neg.inl\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₁' p₂ p₃ : P\nh : Collinear ℝ {p₁, p₂, p₁'}\nhp₁p₂ : p₁ ≠ p₂\nhp₁'p₂ : p₁' ≠ p₂\... | rw [hw'.oangle_eq_add_pi_left hp₃p₂, smul_add, Real.Angle.two_zsmul_coe_pi, add_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 465,
"column": 6
} | {
"line": 465,
"column": 76
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nh : x ≠ 0\nr : ℝ\nhr : 0 < r\n⊢ o.oangle x (r • (o.rotation ↑(π / 2)) x) = ↑(π / 2)",
"usedConstants": [
"LinearIsometryEquiv.instEquivLike",
"Eq... | rw [o.oangle_smul_right_of_pos _ _ hr, o.oangle_rotation_self_right h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 511,
"column": 2
} | {
"line": 518,
"column": 20
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nh : x ≠ 0\nr : ℝ\n⊢ o.oangle (x - r • (o.rotation ↑(π / 2)) x) x = ↑(Real.arctan r)",
"usedConstants": [
"LinearIsometryEquiv.instEquivLike",
"Ad... | by_cases hr : r = 0; · simp [hr]
have hx : x = r⁻¹ • o.rotation (π / 2 : ℝ) (-(r • o.rotation (π / 2 : ℝ) x)) := by
simp [hr, ← Real.Angle.coe_add]
rw [sub_eq_add_neg, add_comm]
nth_rw 3 [hx]
nth_rw 2 [hx]
rw [o.oangle_add_left_smul_rotation_pi_div_two, inv_inv]
simpa [hr] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 511,
"column": 2
} | {
"line": 518,
"column": 20
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nh : x ≠ 0\nr : ℝ\n⊢ o.oangle (x - r • (o.rotation ↑(π / 2)) x) x = ↑(Real.arctan r)",
"usedConstants": [
"LinearIsometryEquiv.instEquivLike",
"Ad... | by_cases hr : r = 0; · simp [hr]
have hx : x = r⁻¹ • o.rotation (π / 2 : ℝ) (-(r • o.rotation (π / 2 : ℝ) x)) := by
simp [hr, ← Real.Angle.coe_add]
rw [sub_eq_add_neg, add_comm]
nth_rw 3 [hx]
nth_rw 2 [hx]
rw [o.oangle_add_left_smul_rotation_pi_div_two, inv_inv]
simpa [hr] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 717,
"column": 4
} | {
"line": 717,
"column": 85
} | [
{
"pp": "case neg.refine_1\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : ¬x = 0\nhy : ¬y = 0\nh : o.oangle x y = ↑(π / 2) ∨ o.oangle x y = ↑(-π / 2)\n⊢ InnerProductGeometry.angle x y = π / 2",
"usedCon... | rwa [o.angle_eq_abs_oangle_toReal hx hy, Real.Angle.abs_toReal_eq_pi_div_two_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 717,
"column": 4
} | {
"line": 717,
"column": 85
} | [
{
"pp": "case neg.refine_1\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : ¬x = 0\nhy : ¬y = 0\nh : o.oangle x y = ↑(π / 2) ∨ o.oangle x y = ↑(-π / 2)\n⊢ InnerProductGeometry.angle x y = π / 2",
"usedCon... | rwa [o.angle_eq_abs_oangle_toReal hx hy, Real.Angle.abs_toReal_eq_pi_div_two_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 717,
"column": 4
} | {
"line": 717,
"column": 85
} | [
{
"pp": "case neg.refine_1\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : ¬x = 0\nhy : ¬y = 0\nh : o.oangle x y = ↑(π / 2) ∨ o.oangle x y = ↑(-π / 2)\n⊢ InnerProductGeometry.angle x y = π / 2",
"usedCon... | rwa [o.angle_eq_abs_oangle_toReal hx hy, Real.Angle.abs_toReal_eq_pi_div_two_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 674,
"column": 2
} | {
"line": 675,
"column": 94
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nh : ∡ p₁ p₂ p₃ = ↑(π / 2)\nhs : (∡ p₂ p₃ p₁).sign = 1\n⊢ (∡ p₂ p₃ p₁).sin * dist p₁ ... | rw [oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
sin_angle_mul_dist_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 90,
"column": 2
} | {
"line": 92,
"column": 41
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nc p₁ p₂ : P\n⊢ c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = dist p₁ p₂ ^ 2 / 2",
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.... | rw [mem_perpBisector_iff_inner_eq_zero, ← vsub_sub_vsub_cancel_right _ _ p₁, inner_sub_left,
sub_eq_zero, midpoint_vsub_left, invOf_eq_inv, real_inner_smul_left, real_inner_self_eq_norm_sq,
dist_eq_norm_vsub' V, div_eq_inv_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 90,
"column": 2
} | {
"line": 92,
"column": 41
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nc p₁ p₂ : P\n⊢ c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = dist p₁ p₂ ^ 2 / 2",
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.... | rw [mem_perpBisector_iff_inner_eq_zero, ← vsub_sub_vsub_cancel_right _ _ p₁, inner_sub_left,
sub_eq_zero, midpoint_vsub_left, invOf_eq_inv, real_inner_smul_left, real_inner_self_eq_norm_sq,
dist_eq_norm_vsub' V, div_eq_inv_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 90,
"column": 2
} | {
"line": 92,
"column": 41
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nc p₁ p₂ : P\n⊢ c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = dist p₁ p₂ ^ 2 / 2",
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.... | rw [mem_perpBisector_iff_inner_eq_zero, ← vsub_sub_vsub_cancel_right _ _ p₁, inner_sub_left,
sub_eq_zero, midpoint_vsub_left, invOf_eq_inv, real_inner_smul_left, real_inner_self_eq_norm_sq,
dist_eq_norm_vsub' V, div_eq_inv_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 40
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nh : ⟪v, q -ᵥ p⟫ = 0\n⊢ ((signedDist v) p) q = 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSemin... | simpa using signedDist_right_congr p h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 40
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nh : ⟪v, q -ᵥ p⟫ = 0\n⊢ ((signedDist v) p) q = 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSemin... | simpa using signedDist_right_congr p h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 40
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nh : ⟪v, q -ᵥ p⟫ = 0\n⊢ ((signedDist v) p) q = 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSemin... | simpa using signedDist_right_congr p h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Basic | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 16
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nι₁ : Type u_3\ns₁ : Finset ι₁\nw₁ : ι₁ → ℝ\np₁ : ι₁ → P\nh₁ : ∑ i ∈ s₁, w₁ i = 0\nι₂ : Type u_4\ns₂ : Finset ι₂\nw₂ : ι₂ → ℝ\np₂ : ι₂ → P\nh₂ : ∑ i ∈ s₂, w₂ i ... | rcongr (i₁ i₂) | Batteries.Tactic._aux_Batteries_Tactic_Congr___elabRules_Batteries_Tactic_rcongr_1 | Batteries.Tactic.rcongr |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 393,
"column": 4
} | {
"line": 393,
"column": 15
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\np : Fin 3 → P\nhps : Set.range p ⊆ s\nhpi : Function.Injective p\nv : V\nhv0 : v ≠ 0\nc : P\nr : ℝ\nhs : ∀ p ∈ s, dist p c = r\nhs' : ∀ (i : Fin 3),... | exact hs' i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 34
} | [
{
"pp": "case refine_1.inl\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np q : P\nh : s.orthRadius p ≤ s.orthRadius q\nh' : (ℝ ∙ (p -ᵥ s.center))ᗮ ≤ (ℝ ∙ (q -ᵥ s.center))ᗮ\nr : ℝ\nhr : r • (p -ᵥ s.ce... | nth_rw 1 [← one_mul r] at hp | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 71
} | [
{
"pp": "case refine_1.inr\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np q : P\nh : s.orthRadius p ≤ s.orthRadius q\nh' : (ℝ ∙ (p -ᵥ s.center))ᗮ ≤ (ℝ ∙ (q -ᵥ s.center))ᗮ\nr : ℝ\nhr : r • (p -ᵥ s.ce... | rw [hp, vsub_self, smul_zero, eq_comm, vsub_eq_zero_iff_eq] at hr | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 264,
"column": 10
} | {
"line": 264,
"column": 45
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np : P\nh : dist p s.center < s.radius\nhp : p ≠ s.center\nhb : ℝ ∙ (p -ᵥ s.center) ≠ ⊤\n⊢ ¬Subsingleton V",
"usedConstants": [
"Eq.mpr"... | rw [AddTorsor.subsingleton_iff V P] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 465,
"column": 6
} | {
"line": 488,
"column": 27
} | [
{
"pp": "case neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : Nontrivial V\ns₁ s₂ : Sphere P\nh : dist s₁.center s₂.center = s₂.radius - s₁.radius\nh₁ : 0 ≤ s₁.radius\nh₂ : 0 ≤ s₂.radius\nh0 : ¬s₁.cente... | rw [dist_comm] at h
have ha : |s₂.radius - s₁.radius| = s₂.radius - s₁.radius := by
refine abs_of_nonneg ?_
rw [← h]
exact dist_nonneg
have hr0 : s₂.radius - s₁.radius ≠ 0 := by
intro hr0
rw [hr0, dist_eq_zero] at h
exact h0 h.symm
refine ⟨AffineMap.line... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 465,
"column": 6
} | {
"line": 488,
"column": 27
} | [
{
"pp": "case neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : Nontrivial V\ns₁ s₂ : Sphere P\nh : dist s₁.center s₂.center = s₂.radius - s₁.radius\nh₁ : 0 ≤ s₁.radius\nh₂ : 0 ≤ s₂.radius\nh0 : ¬s₁.cente... | rw [dist_comm] at h
have ha : |s₂.radius - s₁.radius| = s₂.radius - s₁.radius := by
refine abs_of_nonneg ?_
rw [← h]
exact dist_nonneg
have hr0 : s₂.radius - s₁.radius ≠ 0 := by
intro hr0
rw [hr0, dist_eq_zero] at h
exact h0 h.symm
refine ⟨AffineMap.line... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Bisector | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 82
} | [
{
"pp": "case pos\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np p₁ p₂ p₃ : P\nh₂ : p₁ ≠ p₂\nh₃ : p₁ ≠ p₃\nh : 2 • ∡ p₂ p₁ p = 2 • ∡ p p₁ p₃\nh' : ↑... | exact dist_orthogonalProjection_line_eq_of_two_zsmul_oangle_eq_aux₂ h₂ h₃ h h' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Angle.Bisector | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 82
} | [
{
"pp": "case pos\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np p₁ p₂ p₃ : P\nh₂ : p₁ ≠ p₂\nh₃ : p₁ ≠ p₃\nh : 2 • ∡ p₂ p₁ p = 2 • ∡ p p₁ p₃\nh' : ↑... | exact dist_orthogonalProjection_line_eq_of_two_zsmul_oangle_eq_aux₂ h₂ h₃ h h' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Bisector | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 82
} | [
{
"pp": "case pos\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np p₁ p₂ p₃ : P\nh₂ : p₁ ≠ p₂\nh₃ : p₁ ≠ p₃\nh : 2 • ∡ p₂ p₁ p = 2 • ∡ p p₁ p₃\nh' : ↑... | exact dist_orthogonalProjection_line_eq_of_two_zsmul_oangle_eq_aux₂ h₂ h₃ h h' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 353,
"column": 2
} | {
"line": 355,
"column": 44
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nhf2 : Fact (Module.finrank ℝ V = 2)\ns : Sphere P\np : P\nhp : dist p s.center ≤ s.radius\nhpc : p ≠ s.center\nv : V\nhv : v ∈ (ℝ ∙ (p -ᵥ s.center))ᗮ\nhv0 : v ... | convert!
inter_orthRadius_eq_of_dist_le_radius_of_norm_eq_one hp hpc (v := ‖v‖⁻¹ • v)
(Submodule.smul_mem _ _ hv) ?_ using 2 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Geometry.Euclidean.Angle.Incenter | {
"line": 194,
"column": 4
} | {
"line": 195,
"column": 22
} | [
{
"pp": "case inr.inr.inr\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\... | rw [hs, oangle_rev (t.points i₃), t.oangle_excenter_singleton_eq_add_pi h₁₃.symm h₂₃.symm h₁₂,
oangle_rev] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Unoriented.CrossProduct | {
"line": 34,
"column": 2
} | {
"line": 35,
"column": 80
} | [
{
"pp": "a b : EuclideanSpace ℝ (Fin 3)\nthis : 0 ≤ sin (angle a b)\n⊢ ‖toLp 2 ((crossProduct a.ofLp) b.ofLp)‖ ^ 2 = ‖a‖ ^ 2 * ‖b‖ ^ 2 - ⟪a, b⟫ ^ 2",
"usedConstants": [
"WithLp",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"PiLp.instNorm",
"Norm.norm",
"Eq.mpr"... | · simp_rw [norm_sq_eq_re_inner (𝕜 := ℝ), EuclideanSpace.inner_eq_star_dotProduct, star_trivial,
RCLike.re_to_real, cross_dot_cross, dotProduct_comm (ofLp b) (ofLp a), sq] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 253,
"column": 12
} | {
"line": 253,
"column": 70
} | [
{
"pp": "V : Type u_3\nP : Type u_4\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\ns : Sphere P\np₁ p₂ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nh : p₁ ≠ p₂\nr : ℝ\nhr : r • («o».rotation ↑(... | show p₂ -ᵥ p₁ = (2 : ℝ) • (midpoint ℝ p₁ p₂ -ᵥ p₁) by simp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 31
} | [
{
"pp": "case refine_2\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns : Finset (Fin (n + 1))\ni : Fin (n + 1)\n⊢ (s.height i)⁻¹ ≠ 0",
"usedConstants": [
... | simp [(s.height_pos i).ne'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 31
} | [
{
"pp": "case refine_2\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns : Finset (Fin (n + 1))\ni : Fin (n + 1)\n⊢ (s.height i)⁻¹ ≠ 0",
"usedConstants": [
... | simp [(s.height_pos i).ne'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 31
} | [
{
"pp": "case refine_2\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns : Finset (Fin (n + 1))\ni : Fin (n + 1)\n⊢ (s.height i)⁻¹ ≠ 0",
"usedConstants": [
... | simp [(s.height_pos i).ne'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 512,
"column": 8
} | {
"line": 512,
"column": 29
} | [
{
"pp": "case neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ : P\nh : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄\nhc : Collinear ℝ {p₁, p₂, p₄}\nhe ... | Set.insert_comm p₁ p₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 22
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhpi : angle x y ≠ π\nh : ⟪x, x - y⟫ / (‖x‖ * ‖x - y‖) = ⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)\n⊢ ‖x‖ = ‖y‖",
"usedConstants": [
"Norm.norm",
"Real",
"Classical.propDecidable",
"dite",
"NormedAd... | by_cases hxy : x = y | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 554,
"column": 16
} | {
"line": 554,
"column": 58
} | [
{
"pp": "case h.e'_2.a.pointIndex\nn : ℕ\na✝¹ : Fin (n + 1)\na✝ : pointIndex a✝¹ ∈ univ\n⊢ circumcenterWeightsWithCircumcenter n (pointIndex a✝¹) =\n if pointIndex a✝¹ = circumcenterIndex then Function.const (PointsWithCircumcenterIndex n) 1 (pointIndex a✝¹) else 0",
"usedConstants": [
"False",
... | simp [circumcenterWeightsWithCircumcenter] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 554,
"column": 16
} | {
"line": 554,
"column": 58
} | [
{
"pp": "case h.e'_2.a.circumcenterIndex\nn : ℕ\na✝ : circumcenterIndex ∈ univ\n⊢ circumcenterWeightsWithCircumcenter n circumcenterIndex =\n if circumcenterIndex = circumcenterIndex then Function.const (PointsWithCircumcenterIndex n) 1 circumcenterIndex\n else 0",
"usedConstants": [
"Real",
... | simp [circumcenterWeightsWithCircumcenter] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 98
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ ⟪y, NormedSpace.normalize (ortho y x)⟫ = 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero"... | simp only [NormedSpace.normalize, real_inner_smul_right, mul_eq_zero, inv_eq_zero, norm_eq_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 561,
"column": 2
} | {
"line": 561,
"column": 29
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni : Fin (n + 1)\n⊢ (∑ x, (if x ∈ signs then -1 else 1) *... | simp [(s.height_pos i).ne'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 348,
"column": 2
} | {
"line": 358,
"column": 32
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c p : P\nhb : a ≠ b\nhc : a ≠ c\nhp : Wbtw ℝ b p c\n⊢ ∠ b a p + ∠ p a c = ∠ b a c",
"usedConstants": [
"Iff.mpr",
"EuclideanGeometry.angle_... | by_cases pb : p = b; · simpa [pb] using angle_self_of_ne hb.symm
by_cases pc : p = c; · simpa [pc] using angle_self_of_ne hc.symm
have ea := angle_add_angle_add_angle_eq_pi c hb
have eb := angle_add_angle_add_angle_eq_pi p hb
have ec := angle_add_angle_add_angle_eq_pi p hc.symm
replace hp : ∠ b p c = π := ang... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 348,
"column": 2
} | {
"line": 358,
"column": 32
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c p : P\nhb : a ≠ b\nhc : a ≠ c\nhp : Wbtw ℝ b p c\n⊢ ∠ b a p + ∠ p a c = ∠ b a c",
"usedConstants": [
"Iff.mpr",
"EuclideanGeometry.angle_... | by_cases pb : p = b; · simpa [pb] using angle_self_of_ne hb.symm
by_cases pc : p = c; · simpa [pc] using angle_self_of_ne hc.symm
have ea := angle_add_angle_add_angle_eq_pi c hb
have eb := angle_add_angle_add_angle_eq_pi p hb
have ec := angle_add_angle_add_angle_eq_pi p hc.symm
replace hp : ∠ b p c = π := ang... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 769,
"column": 2
} | {
"line": 770,
"column": 48
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\nn : ℕ\ninst✝¹ : NeZero n\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\nh : i ≠ j\n⊢ (affineSpan ℝ (Set.range (s.faceOpposite i).points)).SSameS... | rw [(s.excenterExists_singleton j).sSameSide_excenter_point_iff, ← sign_eq_one_iff,
s.sign_excenterWeights_singleton_pos h.symm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 769,
"column": 2
} | {
"line": 770,
"column": 48
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\nn : ℕ\ninst✝¹ : NeZero n\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\nh : i ≠ j\n⊢ (affineSpan ℝ (Set.range (s.faceOpposite i).points)).SSameS... | rw [(s.excenterExists_singleton j).sSameSide_excenter_point_iff, ← sign_eq_one_iff,
s.sign_excenterWeights_singleton_pos h.symm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 769,
"column": 2
} | {
"line": 770,
"column": 48
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\nn : ℕ\ninst✝¹ : NeZero n\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\nh : i ≠ j\n⊢ (affineSpan ℝ (Set.range (s.faceOpposite i).points)).SSameS... | rw [(s.excenterExists_singleton j).sSameSide_excenter_point_iff, ← sign_eq_one_iff,
s.sign_excenterWeights_singleton_pos h.symm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 847,
"column": 2
} | {
"line": 847,
"column": 81
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni : Fin (n + 1)\n⊢ |↑(SignType.sign (∑ j, s.excenterWeig... | rcases lt_trichotomy 0 (∑ i, s.excenterWeightsUnnorm signs i) with h' | h' | h' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Geometry.Euclidean.Inversion.Calculus | {
"line": 95,
"column": 2
} | {
"line": 101,
"column": 32
} | [
{
"pp": "F : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nc : F\nR : ℝ\nx : F\nhx : (fun x ↦ c + x) x ≠ c\n⊢ HasFDerivAt (inversion c R) ((R / dist ((fun x ↦ c + x) x) c) ^ 2 • ↑↑(ℝ ∙ ((fun x ↦ c + x) x - c))ᗮ.reflection)\n ((fun x ↦ c + x) x)",
"usedConstants": [
"Contin... | have : HasFDerivAt (inversion c R) (?_ : F →L[ℝ] F) (c + x) := by
simp +unfoldPartialApp only [inversion]
simp_rw [dist_eq_norm, div_pow, div_eq_mul_inv]
have A := (hasFDerivAt_id (𝕜 := ℝ) (c + x)).sub_const c
have B := ((hasDerivAt_inv <| by simpa using hx).comp_hasFDerivAt _ A.norm_sq).const_mul
... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1127,
"column": 4
} | {
"line": 1128,
"column": 8
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\nm n : ℕ\ninst✝¹ : NeZero m\ninst✝ : NeZero n\ns : Simplex ℝ P n\ne : Fin (n + 1) ≃ Fin (m + 1)\nsigns : Finset (Fin (m + 1))\ni : Fin (m + 1)\n⊢ ∑ j, (s.touch... | rw [Finset.sum_comp_equiv]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1127,
"column": 4
} | {
"line": 1128,
"column": 8
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\nm n : ℕ\ninst✝¹ : NeZero m\ninst✝ : NeZero n\ns : Simplex ℝ P n\ne : Fin (n + 1) ≃ Fin (m + 1)\nsigns : Finset (Fin (m + 1))\ni : Fin (m + 1)\n⊢ ∑ j, (s.touch... | rw [Finset.sum_comp_equiv]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Similarity | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 79
} | [
{
"pp": "case h.h\nV₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTors... | exact similar_of_side_side (by positivity) (by positivity) h_sin2 h_sin1.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 92,
"column": 27
} | {
"line": 92,
"column": 60
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nm n : ℕ\ns : Simplex ℝ P n\ne : Fin (n + 1) ≃ Fin (m + 1)\n⊢ n = m",
"usedConstants": [
"Iff.mpr",
"Fintype.card_fin",
"congrArg",
... | simpa using Fintype.card_eq.2 ⟨e⟩ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 92,
"column": 27
} | {
"line": 92,
"column": 60
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nm n : ℕ\ns : Simplex ℝ P n\ne : Fin (n + 1) ≃ Fin (m + 1)\n⊢ n = m",
"usedConstants": [
"Iff.mpr",
"Fintype.card_fin",
"congrArg",
... | simpa using Fintype.card_eq.2 ⟨e⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 92,
"column": 27
} | {
"line": 92,
"column": 60
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nm n : ℕ\ns : Simplex ℝ P n\ne : Fin (n + 1) ≃ Fin (m + 1)\n⊢ n = m",
"usedConstants": [
"Iff.mpr",
"Fintype.card_fin",
"congrArg",
... | simpa using Fintype.card_eq.2 ⟨e⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Similarity | {
"line": 103,
"column": 70
} | {
"line": 103,
"column": 84
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\n... | dist_comm a _, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 234,
"column": 2
} | {
"line": 235,
"column": 81
} | [
{
"pp": "case neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P (n + 2)\ni₁ i₂ : Fin (n + 3)\nh : ¬i₁ = i₂\n⊢ ⟪s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points, s.points i₁ -ᵥ s.points i... | simp_rw [mongePoint_vsub_face_centroid_eq_weightedVSub_of_pointsWithCircumcenter s h,
point_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Euclidean.Sphere.Ptolemy | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 68
} | [
{
"pp": "case h\nV : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_2\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : ∠ a p c = π\nhbpd : ∠ b p d = π\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b... | rwa [angle_comm, angle_eq_angle_of_angle_eq_pi_of_angle_eq_pi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 279,
"column": 27
} | {
"line": 279,
"column": 60
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nm n : ℕ\ns : Simplex ℝ P (n + 2)\ne : Fin (n + 3) ≃ Fin (m + 3)\ni₁ i₂ : Fin (m + 3)\n⊢ n = m",
"usedConstants": [
"Iff.mpr",
"Fintype.card_fin... | simpa using Fintype.card_eq.2 ⟨e⟩ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 279,
"column": 27
} | {
"line": 279,
"column": 60
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nm n : ℕ\ns : Simplex ℝ P (n + 2)\ne : Fin (n + 3) ≃ Fin (m + 3)\ni₁ i₂ : Fin (m + 3)\n⊢ n = m",
"usedConstants": [
"Iff.mpr",
"Fintype.card_fin... | simpa using Fintype.card_eq.2 ⟨e⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 279,
"column": 27
} | {
"line": 279,
"column": 60
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nm n : ℕ\ns : Simplex ℝ P (n + 2)\ne : Fin (n + 3) ≃ Fin (m + 3)\ni₁ i₂ : Fin (m + 3)\n⊢ n = m",
"usedConstants": [
"Iff.mpr",
"Fintype.card_fin... | simpa using Fintype.card_eq.2 ⟨e⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 329,
"column": 34
} | {
"line": 329,
"column": 94
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P (n + 2)\ni₁ : Fin (n + 3)\np : P\nh : ∀ (i₂ : Fin (n + 3)), i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂\ni₂ : Fin (n + 3)\nhne : i₁ ≠ i₂\n⊢ p -ᵥ s.... | vsub_right_mem_direction_iff_mem s.mongePoint_mem_mongePlane | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Group.Growth.LinearLowerBound | {
"line": 36,
"column": 2
} | {
"line": 44,
"column": 51
} | [
{
"pp": "case inr.inr\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nX : Finset G\nn : ℕ\nhX₁ : 1 ∈ X\nhX : X.Nontrivial\nhn : n ≠ 0\nhXn : X ^ n = X ^ (n + 1)\nthis :\n ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] {X : Finset G} {n : ℕ},\n 1 ∈ X → X.Nontrivial → n ≠ 0 → X ^ n = X ^ (... | · simp +contextual only [pow_one] at this
replace hXn d : X ^ (n + d) = X ^ n := by
induction d with
| zero => rw [add_zero]
| succ d hd =>
rw [pow_add, pow_one] at hXn
rw [← add_assoc, pow_add, pow_one, hd, ← hXn]
exact mod_cast this (one_mem_pow hX₁) (hX.pow hn) one_ne_zero
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 368,
"column": 41
} | {
"line": 368,
"column": 67
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nm : ℝ≥0∞ → ℝ≥0∞\nc : ℝ≥0∞\nhc : c ≠ ∞\nhc' : c ≠ 0\n⊢ ⨆ r, ⨆ (_ : r > 0), boundedBy (extend fun s x ↦ (c • m) (ediam s)) =\n ⨆ i, ⨆ (_ : i > 0), boundedBy (c • extend fun s x ↦ m (ediam s))",
"usedConstants": [
"MeasureTheory.ennreal_smul_extend",
... | ennreal_smul_extend _ hc', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 480,
"column": 8
} | {
"line": 480,
"column": 19
} | [
{
"pp": "case pos\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nm : ℝ≥0∞ → ℝ≥0∞\ns : Set X\nr : ℝ≥0∞\nx✝ : r > 0\nt : ℕ → Set X\nht : s ⊆ iUnion t\nhtr : ∀ (n : ℕ), ediam (t n) ≤ r\n⊢ ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ⨅ (_ : ediam (t n) ≤ r), m (ediam (t n)) =\n ... | iInf_eq_if, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 483,
"column": 8
} | {
"line": 483,
"column": 19
} | [
{
"pp": "case neg\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nm : ℝ≥0∞ → ℝ≥0∞\ns : Set X\nr : ℝ≥0∞\nx✝ : r > 0\nt : ℕ → Set X\nht : s ⊆ iUnion t\nhtr : ¬∀ (n : ℕ), ediam (t n) ≤ r\n⊢ ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ⨅ (_ : ediam (t n) ≤ r), m (ediam (t n)) =\n ... | iInf_eq_if, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 486,
"column": 28
} | {
"line": 486,
"column": 39
} | [
{
"pp": "case neg\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nm : ℝ≥0∞ → ℝ≥0∞\ns : Set X\nr : ℝ≥0∞\nx✝ : r > 0\nt : ℕ → Set X\nht : s ⊆ iUnion t\nn : ℕ\nhn : r < ediam (t n)\n⊢ ⨅ (_ : ediam (t n) ≤ r), m (ediam (t n)) = ∞",
"usedConstants": [
"Eq.mpr",
... | iInf_eq_if, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 638,
"column": 2
} | {
"line": 638,
"column": 23
} | [
{
"pp": "X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\ns : Set X\nh : s.Nonempty\n⊢ 1 ≤ μH[0] s",
"usedConstants": []
}
] | rcases h with ⟨x, hx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
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